Problem source

\begin{questionparts}
\item Prove that the intersection of the surface
of a sphere with a plane is always a circle, a point or the empty
set. Prove that the intersection of the surfaces of two spheres with
distinct centres is always a circle, a point or the empty set. 

{[}If you use coordinate geometry, a careful choice of origin and
axes may help.{]} 

\item The parish council of Little Fitton have just bought a modern
sculpture entitled `Truth, Love and Justice pouring forth their blessings
on Little Fitton.' It consists of three vertical poles $AD,BE$ and
$CF$ of heights 2 metres, 3 metres and 4 metres respectively. Show
that $\angle DEF=\cos^{-1}\frac{1}{5}.$

Vandals now shift the pole $AD$ so that $A$ is unchanged and the
pole is still straight but $D$ is vertically above $AB$ with $\angle BAD=\frac{1}{4}\pi$
(in radians). Find the new angle $\angle DEF$ in radians correct
to four figures. 
\end{questionparts}